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Evaluation of Definite Integrals by Substitution - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JMI B.Tech Admission

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  • 19 Questions around this concept.

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The solution for x of the equation  \int_{\sqrt{2}}^{x}\; \frac{dt}{t\sqrt{t^{2}-1}}=\frac{\pi }{2}     is

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Concepts Covered - 1

Evaluation of Definite Integrals by Substitution

We have already learnt to find Indefinite Integration by using substitution method. But in the case of definite integration, we also need to change the limits of integration 'a' and 'b'. If we substitute x = g(t), then g(t)  must be continuous in the interval [a, b].

Let's see some solved examples to see how such questions are solved.

Example 1

\text{Compute the integral } \;\int_{0}^{\pi / 2} \frac{d x}{a^{2} \cos ^{2} x+b^{2} \sin ^{2} x}

\\\text { Let } \quad I=\int_{x=0}^{x=\pi / 2} \frac{d x}{a^{2} \cos ^{2} x+b^{2} \sin ^{2} x}\\\text{Divide numerator and denominator by }\cos^2x\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;}=\int_{x=0}^{x=\pi / 2} \frac{\sec ^{2} x d x}{a^{2}+b^{2} \tan ^{2} x}\\\text { Put } \quad \text { tan } x=t \Rightarrow \sec ^{2} x d x=d t\\\therefore\;\;\;\; \quad I=\int_{t=0}^{t=\infty} \frac{d t}{a^{2}+b^{2} t^{2}}\\\begin{array}{l}{\text { We find the new limits of integration } t=\tan x \Rightarrow t=0} \\ {\text { when } x=0 \text { and } t=\infty \text { when } x=\pi / 2}\end{array}\\\mathrm{\Rightarrow\;\;\;\;} \quad I=\frac{1}{b^{2}} \int_{0}^{\infty} \frac{d t}{\left(\frac{a}{b}\right)^{2}+t^{2}}=\frac{1}{b^{2}} \cdot \frac{1}{a / b}\left[\tan ^{-1} \frac{b t}{a}\right]_{0}^{\infty}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;}=\frac{1}{a b}\left[\frac{\pi}{2}-0\right]=\frac{\pi}{2 a b}

 

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Evaluation of Definite Integrals by Substitution

Study other Related Concepts

Evaluation of Definite Integrals by Substitution Current Topic

Integration as an Inverse Process of Differentiation
Integration of Trigonometric Functions
Integration by Substitution
Fundamental Formulae of Indefinite Integration (Inverse Trigonometric Functions)
Integrals of Particular Function
Trigonometric Integrals
Integration By Parts Formula
Integration Using Partial Fraction
Integration of Irrational Algebraic Function
Reduction Formula

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